Skip to main content
No. 3071:
The Unreasonable Effectiveness of Mathematics

by Krešimir Josić

Today, the unreasonable effectiveness of mathematics. The University of Houston presents this series about the machines that make our civilization run, and the people whose ingenuity created them.

The 16th century astronomer Galileo Galilei claimed that the language of nature is mathematics. As far as we can see, Galileo was right. But why is math the language of nature? And why is that language understandable to us? 

These questions puzzled Eugene Wigner, a physicist who was part of the revolution that brought us quantum mechanics and relativity. Wigner and his colleagues used math to express the laws of nature. To their surprise, they found that mathematicians had already invented the language they needed. They'd developed this language without thinking of whether or how physicists or anybody else would apply their ideas.

This made Wigner ask why math is so unreasonably effective in describing nature. Much is hidden behind this question: First, why do general laws of nature exist? Even if we assume that without such laws all would be chaos, it is still a wonder that we can discover and understand them. But let us accept that nature is humanly comprehensible.

Photo of Heisenberg (left) handing a pen to Wigner (right). Photo Credit: Wikimedia Commons

What still puzzled Wigner is that we have a language of our own making ready at hand to describe the world around us. The words, phrases and ideas of mathematics we need to write the laws of nature are often available when we need them. In math ideas are developed because they naturally flow from previous theories, and because mathematicians find them beautiful. It is then somewhat of a miracle that some of these ideas can be applied not just in physics, but in many other sciences.

Some have argued that math is not miraculously useful. We might be focusing on problems where math happens to be of great help. In medicine, economics, and the social sciences general mathematical laws have been harder to come by.

Some data scientists have therefore argued that we need to embrace the complexity of such systems. They say we should replace elegant mathematics with pragmatic approaches, and let the data guide us. Yet in practice successful data scientists still rely on math.

Mathematics may not be able to unlock all mysteries. But I do side with those that see it as unreasonably useful — it lets us describe and understand things infinitesimally small, unimaginably large, and events far in the past and the future. Math gives us a glimpse into realms that we can't directly experience, and where our intuitions are of no use.

Eugene Wigner concluded that mathematics is a wonderful gift which we neither understand, nor deserve. He expressed a hope that it will continue to be useful to our continued surprise. I do believe that we will increasingly rely on computers to help us make sense of the world around us. But I am sure that math will remain the language in which we will express our understanding. 

(Theme music)

Wigner's original essay can be found here

Although over 50 years old, it asks questions that we are not much closer to answering. There have been many follow-ups to Wigner's essay. I have drawn from the ideas of the mathematician R. W. Hamming, and the engineer Derek Abbott .

The precise quote from Galileo "[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word." 

I thank Prof. Alan Huckleberry for emphasizing that there is a natural flow of ideas and developments in mathematics that become apparent in retrospect. Yet, I would like to add that there is also a cultural component to these developments. It may thus be better to view the co-development of physics and mathematics as part of a larger cultural evolution of ideas. Perhaps in this larger context the relation between mathematics and physics is more natural. I leave this discussion to historians of science. 

This episode was first aired on June 21, 2016